analysis of the sub-wavelength grating in optifdtd … of subwavelength...analysis of the...
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Analysis of the sub-wavelength grating in OptiFDTD simulator
Prepared by:
Dileep Dhakal
Masters of Science in Nanomolecular Science
Submitted to:
Prof. Dr. Dietmar Knipp
December 2009
School of Engineering and Science
2
ACKNOWLEDGEMENT
I express my cordial and humble gratitude to Prof. Dr. Dietmar Knipp for giving me
an opportunity to carry out profound research in sub wavelength gratings. I would also like to
thank him for his continuous support. I thank Mr. Vladislav Jovan Jovanov for his constant
guidance, without him the research wouldn‟t have completed in time.
I would also like to thank Mr. Ujwol Palanchoke for the support for the research
activity.
Dileep Dhakal
3
ABSTRACT
Standard sub-wavelength grating array is designed and 2D simulation is carried out, on
OptiFDTD 8.0 simulation software. Implementation of different types of sub-wavelength
grating are discussed and analyzed with simulation results.
The power transmittance at different wavelength under the variation of different grating
parameters is demonstrated on the simulation. Grating periods, grating spacing, grating
material and the presence and absence of substrate are the parameters varied for the
transmittance analysis and also effects of the parameters on power transmittance are studied.
Finally, profound analysis of the results is done to have better understanding of sub-
wavelength gratings.
4
MOTIVATION
The main purpose behind profound analysis of sub-wavelength gratings is to understand the
effect of grating parameters on the measured transmittance at different wavelength of visible
spectrum. Grating spacing, grating period, substrates and materials are basic properties which
may have profound effect on the measured power transmittance, which must be individually
explored. Also, studying such individual effects might help us in designing high efficiency
sub-wavelength gratings. We can also extract any individual property of sub-wavelength
grating and realize the novel system which cannot be thought of in micro-sized world.
For e.g. if grating which produces maximum transmittance (~100%) at particular wavelength
(say, 620-750nm) then, this grating can be applied for filtering red light. This approach will
help in designing ultra low cost color filters.
5
TABLE OF CONTENTS
Title Page No.
Chapter 1: Introduction 5
Chapter 2: Activities and Results 11
Summary of Results 18
References 19
Appendix: Simulation Results 20
6
Chapter 1: Introduction
Sub-wavelength grating
Gratings where grating period is very small and less than the wavelength (λ) of incident light
are termed as sub-wavelength gratings (SWG). For sufficiently small period higher order
diffraction are suppressed and only remains is the zero order diffraction. Thus, SWG can be
represented as the homogeneous medium with optical properties determined by the grating
geometry. There are lots of applications of sub-wavelength grating like anti-reflecting filters
[1], phase plates [2], plasmonic lens [3] and negative refractive index [4].
FDTD (Finite difference Time Domain)
FDTD was first developed by Kane S. Yee in 1966. FDTD is the modeling technique to solve
the Maxwell equation applying finite difference mathematics. FDTD directly calculates the
value of E (Electric field intensity) and H (magnetic field intensity) at different points of the
computational domain (i, j, k). Yee proposed the mesh (Fig. 1), where magnetic and electrical
field wave vector are distributed such that magnetic field lies exactly half points of the
electrical field.
Fig. 1: Yee‟s mesh in 3D. In the mesh, electrical field points are exactly half grid offset from the
magnetic field points [6].
7
In the 2D TE case (Hx, Ey and Hz are nonzero components, for wave propagating along z-
direction and xz-plane is the point of observation) in lossless media, Maxwell's equations take
the following form [6]:
z
E
t
H yx
0
1
,
x
E
t
H yz
0
1
,
x
H
z
H
t
Ezxy
1.............. (1)
Where, r 0 is the dielectric permittivity and 0 is the magnetic permeability of the
vacuum. The refractive index is rn [6].
Fig. 2: Mesh for FDTD in TE mode, when z axis is the direction of propagation of wave and
y-axis is the point of observation. Hx, Ey and Hz are correspondingly nonzero values of
electrical field along x-direction, electric field along y-direction and magnetic field along z-
direction. Δx and Δz are the unit mesh size along the x and z-direction.
8
Relation of stability with the mesh size:
For high degree of accuracy, the grid spacing (Δx, Δy and Δz) in the simulation must be less
than the wavelength, usually less than λ/10. The stability condition which must be satisfied
and termed as Courant limit [5] is given as:
2
1
222max
111
zyxtv …………. (2)
Where, maxv is the maximum velocity of the input wave. When the step size Δ same in all
directions, the stability condition is: [5]
N
tv 1max
…………. (3)
Where, N is the number of spatial dimensions in the problem.
To model the scattering produced by cells at optical wavelengths, the grid size (Δ) must be
small. If a wavelength of λ = 0.5nm is applied, the maximum step size is:
nm05.010
Finite difference method:
Derivative in the partial differential equation are approximated by linear combinations of
functions at the grid point. If we have a 1-D grid, for grid point from 0 to X, functional value
Fi = F (xi), i = 0, 1,…, N, then,
Mesh size is N
Xx and xixi .
9
Fig. 3: Finite difference approximation plot for the function F(x) [7].
For the finite difference method we can write the following types of equation as [7]:
Forward difference equation:
x
xFxxF
x
xF
x
)()()(lim
0
Equals, x
FF
x
F i
i
1 ………….(4)
Backward difference equation:
x
xxFxF
x
xF
x
)()()(lim
0
Equals, x
FF
x
F i
i
1 ………….(5)
Central difference equation:
x
xxFxxF
x
xF
x
2
)()()(lim
0
Equals, x
FF
x
F ii
i
2
11 ………….(6)
The central difference equation yield more accurate results [7].
Applying finite difference method on the equation (1), where we can directly calculate the
values of Ey, Hx and Hz at different points (i, k) [6].
10
Applying above equations, we can directly calculate the values of Ey, Hx and Hz at different
points (i, k).
Power Transmittance
Power Transmittance (Tλ) measured at certain wavelength (λ) is the ratio of power transmitted
to the power incident by the certain material having width (w).
Fig. 4: Power transmission calculation
in
out
P
PT ……..(10)
weT
Where, is the absorption coefficient and w is the path length.
11
Chapter 2: Activities and Results
Transmittance for different conditions of sub-wavelength grating (size: 100nm) has been
studied on OptiFDTD simulator. Power transmittance for different grating periods (K),
grating spacing (d), different grating materials, grating placed in air, grating placed on the
glass and grating placed inside the glass, is calculated. Grating period (K) and grating spacing
(d) is varied and power transmittance (T) is analyzed at different wavelength. Optical source
applied is Gaussian modulated continuous wave with wavelength of 0.5nm.
Fig. 5: Array of gratings placed in wafer (air).
Fig. 6: Array of gratings placed on the substrate (glass). Detector is inside the substrate.
12
Fig. 7: Array of gratings placed inside the glass substrate. Detector is also inside the substrate.
Variation of transmittance with grating spacing (d):
Transmittance vs Wavelength for Perfect Conductor grating
0
0.2
0.4
0.6
0.8
1
1.2
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce
PC_K400_d320
PC_K400_d240
PC_K400_d200
PC_K400_d160
PC_K400_d80
Fig. 8: Plots depicting the variation of transmittance with variation of grating spacing (d), for grating
period (K = 400nm).
13
Transmittance vs Wavelength for Perfect Conductor grating
0
0.2
0.4
0.6
0.8
1
1.2
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce PC_K500_d400
PC_K500_d300
PC_K500_d250
PC_K500_d200
PC_K500_d100
Fig. 9: Plots depicting the variation of transmittance with variation of grating spacing (d), for grating
period (K = 500nm).
In fig. 8, when spacing (d) is 320nm for period (K) as 400nm i.e. spacing is 80% of the
period, then very high transmittance is found, compared with those cases having slightly low
spacing size. Lowest transmittance is seen for very small spacing (80nm) i.e. spacing is 20%
of the grating period. The same result is seen in the case for K = 500nm and d-varied from
20% to 80% (fig. 9). Both cases showed some exciting results, (a) maximum transmittance is
found at the wavelength equal to the period and (b) for 50% spacing there is uniform
transmittance before the wavelength equal to grating period and later decreases rapidly. For
e.g. in fig. 8, constant transmittance is seen upto wavelength 400nm (same as period), and
later decreased rapidly. Similar result is seen in fig. 9 happening at 500nm (which is equal to
the period).
14
Variation of transmittance with grating period (d):
Fig. 10: Above plot depicts the variation of transmittance with grating spacing (d= 250nm) for
different grating period (K) for Perfect conductor grating, grating in the air.
When grating period is varied, keeping grating spacing constant (250nm), maximum
transmittance is seen for the wavelength at corresponding grating period (K). Maximum
transmittance is shifted by the value equal to grating period (Fig. 10). Here also, when
K=500nm and d=250nm i.e. spacing is 50% of the period, constant transmittance is observed
before the wavelength equal to period and later rapid decrement of transmittance is observed.
15
Variation of transmittance, when grating is placed inside substrate (glass) and outside
the substrate:
Fig. 11: Grating within the glass (Fig. 3). Above plot depicts the variation of transmittance with
grating period (K= 500nm) for different grating spacing (d), for grating lies inside the glass substrate
for Perfect Conducting material as grating.
Transmittance vs Wavelength for Perfect Conductor grating (Case 2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce PC_K500_d400
PC_K500_d300
PC_K500_d250
PC_K500_d200
PC_K500_d100
Fig. 12: Grating on the glass (Fig. 2). Above plot depicts the variation of transmittance with grating
period (K= 500nm) for different grating spacing (d), for grating lies on the glass substrate for Perfect
Conducting material as grating.
16
When grating is placed on the glass (µ=1.5) (Fig. 12), transmittance at different spacing had
different maximum transmittance, but when grating is placed inside the glass substrate then
maximum transmittance for different spacing are pretty similar in values. It is seen as the
coinciding of different peaks of the transmittance vs. wavelength curve (Fig. 11). Studying the
case, where, d is 50% of the period, grating placed on the glass substrate showed similar result
as constant transmittance before the wavelength equal to period, and later decreases rapidly.
But, when grating is placed inside the glass substrate, we can observer filter like property
where, transmittance increases rapidly upto maximum transmittance and decreases rapidly
after attaining maximum value (fig. 13).
Fig. 13: Grating in the glass (Fig. 2). Above plot depicts the variation of transmittance with grating
period (K= 500nm) for 50% grating spacing (d), for grating lies inside the glass substrate for Perfect
Conducting material as grating.
17
Variation of transmittance with grating material:
Aluminium showed maximum transmittance when compared with other materials like: perfect
conductor and copper (fig. 14), for spacing (d) kept 50% of period (K).
Transmittance vs Wavelength (nm) for different Materials
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
nsm
itta
nce
PC_K400_d200
Al_K400_d200
Cu_K400_d200
Fig. 14: Transmittance for different materials for grating in the air. Above plot depicts the variation of
transmittance for different grating material at 400nm grating period (K) with grating spacing (d) of
200nm.
18
Summary of Results
Studying the simulation results, it‟s very clear that, variation of sub-wavelength grating
parameter has strong effect on the measured power transmittance at different wavelength.
Effect is notable for spacing (d) is 50% of period (K). Results depicted that sub-wavelength
gratings with spacing (d) with period (K), have maximum transmittance at particular
wavelength equal to period (exponentially rises and after attaining maximum value, decays
exponentially) for d is 50% of the K, only when grating is placed inside the glass. Also, when
grating is outside the substrate or without the substrate and d is 50% of the K, transmittance is
constant before wavelength equals grating period but exponentially decays later.
Additionally, aluminium is found to be better material compared to perfect conductor and
copper, for obtaining high transmittance, after analyzing the results.
19
References
[1] Y. Kanamori, K. Hane, H. Sai, and H. Yugami, “100 nm period silicon anti-reflection
structures fabricated using a porous alumina membrane mask,” Appl. Phys. Lett., Vol.
78, No. 2, 142–143 (2001).
[2] A. G. Lopezand, H. G. Craighead, “Wave-plate polarizing beam splitter based on a
form-birefringentmultilayer grating,‟ Opt. Lett., Vol. 23, No. 20, 1627–1629 (1998).
[3] L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, „Planar Lenses Based on Nanoscale
Slit Arrays in a Metallic Film‟, Nano Lett. Vol. 9, No. 1, 235-238 (2009).Vol. 9, No. 38
[4] J. B. Pendry, „Negative refraction‘, Cont. Phy., Vol. 45, No. 3, 191-202 (2004).
[5] M. Sadiku, „Numerical Techniques in Electromagnetics‟, CRC Press (1992).
[6] „OptiFDTD, Technical Background and Tutorials‟, Optiwave Systems Inc.
[7] „Finite difference method‟, http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/
lecture4.pdf.
20
Appendix: Simulation Results
Variation of transmittance with variation of grating period (K-nm) and grating spacing
(d-nm):
Transmittance vs Wavelength for Perfect Conductor grating
0
0.2
0.4
0.6
0.8
1
1.2
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce
PC_K400_d320
PC_K400_d240
PC_K400_d200
PC_K400_d160
PC_K400_d80
Fig. I: Grating within air for Perfect conductor. Above plot depicts the variation of transmittance with
grating period (K=400nm) for different grating spacing (d) for Perfect conductor grating.
Transmittance vs Wavelength for Perfect Conductor grating
0
0.2
0.4
0.6
0.8
1
1.2
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce PC_K500_d400
PC_K500_d300
PC_K500_d250
PC_K500_d200
PC_K500_d100
Fig. II: Grating within air for Perfect conductor. Above plot depicts the variation of transmittance with
grating period (K= 500nm) for different grating spacing (d) for Perfect conductor grating.
21
Transmittance vs Wavelength for Perfect Conductor grating
0
0.2
0.4
0.6
0.8
1
1.2
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce PC_K600_d480
PC_K600_d360
PC_K600_d300
PC_K600_d240
PC_K600_d120
Fig. III: Grating within air for Perfect conductor. Above plot depicts the variation of transmittance
with grating period (K= 600nm) for different grating spacing (d) for Perfect conductor grating.
Variation of Transmittance with grating period (K-nm)
a. Variation of Transmittance for different grating period (K)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
nsm
itta
nce K400_d200
K450_d200
K500_d200
K550_d200
K600_d200
Fig. IV: Above plot depicts the variation of transmittance with grating spacing (d= 200nm) for
different grating period (K) for Perfect conductor grating, grating in the air.
22
b. Variation of Transmittance for different grating period (K)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
nsm
itta
nce K400_d250
K450_d250
K500_d250
K550_d250
K600_d250
Fig. V: Above plot depicts the variation of transmittance with grating spacing (d= 250nm) for different
grating period (K) for Perfect conductor grating, grating in the air.
Variation of Transmittance For different Materials (Perfect conductor-PC, Copper-Cu
and Aluminium-Al)
Transmittance vs Wavelength (nm) for different Materials
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
nsm
itta
nce
PC_K500_d250
Al_K500_d250
Cu_K500_d250
Fig. VI: Transmittance for different materials. Above plot depicts the variation of transmittance for
different grating material at 500nm grating period (K) with grating spacing (d) of 250nm, grating in
the air.
23
Transmittance vs Wavelength (nm) for different Materials
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
nsm
itta
nce
PC_K400_d200
Al_K400_d200
Cu_K400_d200
Fig. VII: Transmittance for different materials for grating in the air. Above plot depicts the variation of
transmittance for different grating material at 400nm grating period (K) with grating spacing (d) of
200nm.
Transmittance analysis for grating on the glass substrate and in the air:
Transmittance vs Wavelength for Perfect Conductor grating (Case 2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce PC_K500_d400
PC_K500_d300
PC_K500_d250
PC_K500_d200
PC_K500_d100
Fig. VIII: Grating on glass substrate. Above plot depicts the variation of transmittance for grating on
the glass substrate with grating period (K= 500nm) for different grating spacing (d).
24
Transmittance vs Wavelength for Perfect Conductor grating
0
0.2
0.4
0.6
0.8
1
1.2
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce PC_K500_d400
PC_K500_d300
PC_K500_d250
PC_K500_d200
PC_K500_d100
Fig. IX: Grating in Air. Above plot depicts the variation of transmittance for grating in the air with
grating period (K= 500nm) for different grating spacing (d).
Transmittance analysis for grating within the glass substrate and on the substrate
A. For Perfect Conductor grating:
Transmittance vs Wavelength for Perfect Conductor grating (Case 3)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce PC_K500_d400
PC_K500_d300
PC_K500_d250
PC_K500_d200
PC_K500_d100
Fig. X: Grating within the Glass Substrate for Perfect Conductor Grating. Above plot depicts the
variation of transmittance with grating period (K= 500nm) for different grating spacing (d), for grating
lies inside the glass substrate for Perfect Conducting material as grating.
25
Transmittance vs Wavelength for Perfect Conductor grating (Case 2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce PC_K500_d400
PC_K500_d300
PC_K500_d250
PC_K500_d200
PC_K500_d100
Fig. XI: Grating on the Glass Substrate for Perfect Conductor Grating. Above plot depicts the
variation of transmittance with grating period (K= 500nm) for different grating spacing (d), for grating
lies on the glass substrate for Perfect Conducting material as grating.
B. For Copper Grating
Transmittance vs Wavelength for Copper grating (Case 3)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce Cu_K500_d400
Cu_K500_d300
Cu_K500_d250
Cu_K500_d200
Cu_K500_d100
Fig. XII: Grating within the Glass Substrate for Copper Grating. Above plot depicts the variation of
transmittance with grating period (K= 500nm) for different grating spacing (d), for grating lies inside
the glass substrate for Copper as material for grating.
26
Transmittance vs Wavelength for Copper grating (Case 2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce Cu_K500_d400
Cu_K500_d300
Cu_K500_d250
Cu_K500_d200
Cu_K500_d100
Fig. XIII: Grating on the Glass Substrate for Copper Grating. Above plot depicts the variation of
transmittance with grating period (K= 500nm) for different grating spacing (d), for grating lies on the
glass substrate for Copper as material for grating.
C. For Aluminium grating
Transmittance vs Wavelength for Aluminium grating (Case 3)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce Al_K500_d400
Al_K500_d300
Al_K500_d250
Al_K500_d200
Al_K500_d100
Fig. XIV: Grating within the Glass Substrate for Aluminium Grating. Above plot depicts the variation
of transmittance with grating period (K= 500nm) for different grating spacing (d), for grating lies
inside the glass substrate for Aluminium as material for grating.
27
Transmittance vs Wavelength for Aluminium grating (Case 2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8
Wavelength (nm)
Tra
ns
mit
tan
ce Al_K500_d400
Al_K500_d300
Al_K500_d250
Al_K500_d200
Al_K500_d100
Fig. XV: Grating on the Glass Substrate for Aluminium Grating. Above plot depicts the variation of
transmittance with grating period (K= 500nm) for different grating spacing (d), for grating lies on the
glass substrate for Aluminium as material for grating.